It now must be shown that any binary mixture with these equilibrium distributions satisfies the correct hydrodynamic equations. This is done here using the standard Chapman-Enskog method. A similar expansion approach is employed in reference [38] using a perturbation parameter , the time between collisions. Equations (4.105) and (4.106) can be Taylor expanded:
and
Expanding the distribution functions and the time and space derivatives:
and using equations (4.150) and (4.151) we can perform a Chapman-Enskog expansion. Substituting equation (4.152) into equations (4.150) and (4.151) gives
and
It is still require that the total mass and momentum and the mass of each component are conserved at each site. This is achieved, to second-order, if the zeroth-order expansion of the distribution functions are equal to their equilibrium values and if
To first-order in equation (4.153) is
Summing equation (4.156) over i gives the first-order continuity equation for the total population:
Multiplying equation (4.156) by and summing over i gives the first-order continuity equation for the fluid momentum:
Similarly equation (4.154) to first-order in is
Summing over i gives the first-order convection-diffusion equation
To second-order in equation (4.153) is
The first and third terms sum to zero by equation (4.155) and the fourth and fifth terms also sum to zero as in equations (4.157) and (4.158) respectively. Thus summing equation (4.161) we obtain the second-order continuity equation for the whole fluid:
Combining this with the first-order equation (4.157) and recombining the derivatives we get the continuity equation for the whole fluid
Multiplying equation (4.161) by and summing over i gives
To find expressions for the two summations in equation (4.164) consider first
With and the sum of all terms containing an odd number of s being zero this gives
Next consider equation (4.156) multiplied by and summed over i:
Substituting equations (4.166) and (4.167) into equation (4.164) gives
We are dealing here with terms so, since is also small, we can neglect terms . From the definition of , equations (4.114) and (4.115), we also see that, neglecting higher-order derivatives, we can write [38]. We can now re-write the term in square brackets in equation (4.168)
Using the expression for from equation (4.157) we can combine equations (4.158) and (4.168) to obtain the Navier-Stokes equation
where
To second-order in equation (4.154) is
Summing over i and noting that the first and fourth terms are zero we can write
Multiplying equation (4.159) by and summing over i we find the term :
Substituting this into equation (4.173) we get
where as before we have neglected terms smaller than . Finally we can express as [38] and perform the differentiation on both the terms in brackets:
Replacing the time derivatives with spatial derivatives using equations (4.157), (4.158) and (4.160) and, as before, neglecting terms gives
for the final term in equation (4.175). Combining equation (4.175) with equation (4.160) and neglecting terms of order we obtain the convection-diffusion equation
where
The Navier-Stokes equation for the liquid-gas model follows [63, 38] in the same manner as the Navier-Stokes equation for the binary fluid. The kinematic and bulk viscosities have the same dependence as equation (4.171).